Let $G$ be a finite subgroup of $GL_4(\bm{Q})$. The group $G$ induces an
action on $\bm{Q}(x_1,x_2,x_3,x_4)$, the rational function field of four
variables over $\bm{Q}$. Theorem. The fixed subfield
$\bm{Q}(x_1,x_2,x_3,x_4)^G:=\{f\in\bm{Q}(x_1,x_2,x_3,x_4):\sigma \cdot f=f$ for
any $\sigma\in G\}$ is rational (i.e.\ purely transcendental) over $\bm{Q}$,
except for two groups which are images of faithful representations of $C_8$ and
$C_3\rtimes C_8$ into $GL_4(\bm{Q})$ (both fixed fields for these two
exceptional cases are not rational over $\bm{Q}$).

Let $k$ be an infinite field. The notion of retract $k$-rationality was
introduced by Saltman in the study of Noether's problem and other rationality
problems. We will investigate the retract rationality of a field in this paper.
Theorem 1. Let $k\subset K\subset L$ be fields. If $K$ is retract $k$-rational
and $L$ is retract $K$-rational, then $L$ is retract $k$-rational. Theorem 2.
For any finite group $G$ containing an abelian normal subgroup $H$ such that
$G/H$ is a cyclic group, for any complex representation $G \to GL(V)$, the
fixed field $\bm{C}(V)^G$ is retract $\bm{C}$-rational.

Let $G$ be a finite 2-group and $K$ be a field satisfying that (i)
$\fn{char}K\ne 2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on
the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the
fixed field $K(x,y,z)^G$ is rational (= purely transcendental) over $K$.
Applications of this theorem will be given.

We will show the raitonality of some twisted symmetric group actions.